Is there a way to allow certain Instagram users to bypass the approval process and have their photos be immediately added to my slideshow while approving all others?

Yes. When logged in, on the “Instagram Feed Settings” page choose the option “All photos need to be approved before they join the slideshow”, then click on “All Photos” and choose “Approve User” for the users you want to bypass the approval process. The users you want to bypass the approval process will need to have already used your custom hashtag.